In the power systems analysis field of electrical engineering, a per-unit
system is the expression of system quantities as fractions of a defined
base unit quantity. Calculations are simplified because quantities expressed
as per-unit do not change when they are referred from one side of a transformer
to the other. This can be a pronounced advantage in power system analysis where
large numbers of transformers may be encountered. Moreover, similar types of
apparatus will have the impedances lying within a narrow numerical range when
expressed as a per-unit fraction of the equipment rating, even if the unit size
varies widely. Conversion of per-unit quantities to volts, ohms, or amperes
requires a knowledge of the base that the per-unit quantities were referenced
to. The main idea of a per unit system is to
absorb large difference in absolute values into base relationships. Thus,
representations of elements in the system with per unit values become more
uniform. A per-unit system provides units for;
power, voltage, current, impedance, and admittance. Except impedance and
admittance, any two of these are independent and can be arbitrarily
selected as base values, usually power and voltage. All quantities are
specified as multiples of selected base values. For example, the base power
might be the rated power of a transformer, or perhaps an arbitrarily
selected power which makes power quantities in the system more
convenient. The base voltage might be the nominal voltage of a bus. Different
types of quantities are labeled with the same symbol; it should be clear from
context whether the quantity is a voltage, current, etc.
Per-unit being used in power flow, short circuit evaluation and motor starting
studies, it is important for all power engineers to be familiar with the
concept. Purpose
There are several reasons for using a per-unit system:
Similar apparatus will have similar per-unit impedances and losses expressed
on their own rating, regardless of their absolute size. Because of this, per-unit
data can be checked rapidly for gross errors. A per unit value out of normal
range is worth looking into for potential errors.
Manufacturers usually specify the impedance of apparati in per unit
values. Use of the constant is reduced in
three-phase calculations. Per-unit quantities are the same on
either side of a transformer, independent of voltage level
By normalizing quantities to a common base, both hand and automatic
calculations are simplified. It improves numerical stability of
automatic calculation methods Per unit data representation yields
important information about relative magnitudes.
The per-unit system was developed to make manual analysis of power systems
easier. Although power-system analysis is now done by computer, results are
often expressed as per-unit values on a convenient system-wide base.
As we have already known, the purpose of introducing per-unit system is to
simplify our conversion between different transformers. Hence, I’d like
to illustrate the steps for finding per-unit values for voltage and
impedance. First, let the base power of each end of the transformers becomes the
same. Once we set every S on the same base, we can get base voltage and base
impedance for every transformer easily. The numbers we have till now are all
base on the same unit. Next, substitute the real numbers of impedances and
voltages into the per-unit calculation definition. We can get the answers for
per-unit system. In other case, if we’ve known the per-unit values at first, we
can get the real values by timing base values.
Base quantities Generally base values of power and
voltage are chosen. The base power may be the rating of a single piece of
apparatus such as a motor or generator. If a system is being studied, the base
power is usually chosen as a convenient round number such as 10 MVA or 100 MVA.
The base voltage is chosen as the nominal rated voltage of the system. All
other base quantities are derived from these two base quantities. Once the base
power and the base voltage are chosen, the base current and the base impedance
are determined by the natural laws of electrical circuits. Note the base value
should only be magnitudes, while the per-unit values are phasors. The phase
angles of complex power, voltage, current, impedance etc. are not affected
by the conversion to per unit values. As we have already known, the purpose of
introducing per-unit system is to simplify our conversion between
different transformers. Hence, it is appropriate to illustrate the steps for
finding per-unit values for voltage and impedance. First, let the base power of
each end of a transformer become the same. Once we set every S on the same
base, we can get base voltage and base impedance for every transformer easily.
The numbers we have until now are all based on the same unit. Next, substitute
the real numbers of impedances and voltages into the per-unit calculation
definition. We can get the answers for the per-unit system. In other case, if
we have known the per-unit values at first, we can get the real values by
multiplying by the base values. By convention, we adopt the following
two rules for base quantities: The value of base power is the same for
the entire power system of concern. The ratio of the voltage bases on either
side of a transformer is selected to be the same as the ratio of the transformer
voltage ratings. With these two rules, a per-unit
impedance remains unchanged when referred from one side of a transformer
to the other. This allows us to eliminate ideal transformer from a
transformer model. Relationship between units
The relationship between units in a per-unit system depends on whether the
system is single-phase or three-phase. = Single-phase =
Assuming that the independent base values are power and voltage, we have:
Alternatively, the base value for power may be given in terms of reactive or
apparent power, in which case we have, respectively,
or The rest of the units can be derived
from power and voltage using the equations , , and , being represented
by . We have: = Three-phase =
Power and voltage are specified in the same way as single-phase systems.
However, due to differences in what these terms usually represent in
three-phase systems, the relationships for the derived units are different.
Specifically, power is given as total power, and voltage is line-to-line
voltage. In three-phase systems the equations and also hold. The apparent
power now equals Example of per-unit
As an example of how per-unit is used, consider a three-phase power
transmission system that deals with powers of the order of 500 MW and uses a
nominal voltage of 138 kV for transmission. We arbitrarily select ,
and use the nominal voltage 138 kV as the base voltage . We then have:
If, for example, the actual voltage at one of the buses is measured to be 136
kV, we have: Per-unit system formulas
The following tabulation of per-unit system formulas is adapted from Beeman's
Industrial Power Systems Handbook. In transformers
It can be shown that voltages, currents, and impedances in a per-unit system will
have the same values whether they are referred to primary or secondary of a
transformer. The full load copper loss of a
transformer in per-unit form is equal to the per-unit value of its resistance:
Therefore, it may be more useful to express the resistance in per-unit form
as it also represents the full-load copper loss.
References Beeman, Donald. "Short-Circuit-Current
Calculating Procedures". In Beeman, Donald. Industrial Power Systems
Handbook. McGraw-Hill. pp. see esp. 38–41, 52–55.
Elgerd, Olle I.. "§2.5 Per-Unit Representation of Impedances, Currents,
Voltages and Powers". Electric Energy Systems Theory: An Introduction. Tata
McGraw-Hill. pp. 35–39. ISBN 978-0070192300.
Yuen, Moon H.. "Short Circuit ABC--Learn It in an Hour, Use It Anywhere, Memorize
No Formula". IEEE Trans. on Industry Applications. IA-10: 261–272.
doi:10.1109/TIA.1974.349143. William D., Jr, Stevenson,. Elements of
power system analysis. New York: McGraw-Hill. ISBN 0-07-061285-4.
Weedy, B.M.. Electric power systems. London ; Toronto: J. Wiley. ISBN
0-471-92445-8. Glover, J. Duncan; Sarma, Mulukutla;
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